The logarithm operator is the inverse operation of the exponent operator. The operation has a base represented by the variable and operates on an input expression represented by the variable .
The examples below demonstrate the basic pattern the logarithm operator exhibits when the base is the same for the input numbers.
The logarithm operator is closely related to the exponential and radical operators as shown in the table below. Conceptually, the behavior of each of the operators is framed using the scenario of population growth, where the initial population is represented with a value of , the variable represents growth rate, the variable repesents time elapsed, and the variable represents the population.
Equation  Operator 

Exponent Given the growth rate and the time elapsed, the exponent operator returns the population. 

Logarithm Given the growth rate and the population the logarithm operator returns the time elapsed. 

Radical Given the time elapsed and the population the radical operator returns the growth rate. 
Another way to visualize the behavior of these operators is outlined in this article about exponential trees.
Name  Property 

Product  
Quotient 
The exponentiation operator is a binary operator. The base is an expression or number that is being raised to some exponent. The exponent expression is denoted using superscript text.
The radical operator returns the nth root of the provided expression. The radical operator is an alternative way of writing a fractional exponent.
Taking the logarithm of an number is the inverse operation of exponentiation. The subscript number is the base of the logarithm and the expression is the operand.
This article explores a technique of visualizing the three math operators: expononents, logarithms and radicals using the structure of a tree.